Introduction to and Andy Ruina and Rudra Pratap

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Chapter 0. Detailed Contents Detailed Contents 7 Here is an introduction to kinematic constraint in its simplest context, systems that are constrained to move without rotation in a straight line. In one dimension pulley problems provide the main example. Two and three dimensional problems are covered, such as finding structural support forces in accelerating vehicles and the slowing or incipient capsize of a braking car or bicycle. Angular momentum balance is introduced as a needed tool but without the complexities of rotatioinal kinematics. 12.1 1D motion and pulleys . . . . . . . . . . . . . . . . . . . . . 590 12.2 1D motion w/ 2D & 3D forces . . . . . . . . . . . . . . . . . 601 Box 12.1 Calculation of * H =C and P * H=C . . . . . . . . . . . 603 Problems for Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . 614 13 Circular motion 626 After movement on straight-lines the second important special case of motion is rotation on a circular path. Polar coordinates and base vectors are introduced in this simplest possible context. The key new idea is that not just coordinates, but base vectors, can change with time. The primary applications are pendulums, gear trains, and rotationally accelerating motors or brakes. 13.1 Circular motion kinematics . . . . . . . . . . . . . . . . . . 628 Box 13.1 The motion quantities . . . . . . . . . . . . . . . 632 13.2 Dynamics of particle circular motion . . . . . . . . . . . . . 639 Box 13.2 Other derivations of the pendulum equation . . . 643 13.3 2D rigid-object rotation . . . . . . . . . . . . . . . . . . . . 650 Box 13.3 Rotation is uniquely defined for a rigid object (2D) 651 13.4 2D rigid-object angular velocity . . . . . . . . . . . . . . . . 658 Box 13.4 The fixed Newtonian reference frame F . . . . . 659 Box 13.5 Plato on spinning in circles as motion (or not) . . 660 Box 13.6 Acceleration of a point, using * ! . . . . . . . . . 661 Box 13.7 Angular velocity * ! and the rotation matrix ŒR . 664 13.5 Polar moment of inertia . . . . . . . . . . . . . . . . . . . . 670 Box 13.9 Some examples of 2-D Moment of Inertia . . . . 674 Box 13.8 The perpendicular and parallel axis theorems . . 676 13.6 Dynamics of rigid-object planar circular motion . . . . . . . 681 Box 13.10 Angular momentum and power . . . . . . . . . 685 Problems for Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . 712 14 Planar motion of an object 734 The main goal here is to generate equations of motion for general planar motion of a (planar) rigid object that may roll, slide or be in free flight. Multi-object systems are also considered so long as they do not involve other kinematic constraints between the bodies. Features of the solution that can be obtained from analysis are discussed, as are numerical solutions. 14.1 Rigid object kinematics . . . . . . . . . . . . . . . . . . . . 736 14.2 Mechanics of a rigid-object . . . . . . . . . . . . . . . . . . 752 Box 14.1 2-D mechanics makes sense in a 3-D world . . . 758 Box 14.2 The center-of-mass theorems for 2-D rigid bodies 759
8 Chapter 0. Detailed Contents Detailed Contents Box 14.3 The work of a moving force and of a couple . . . 760 Box 14.4 The vector triple product * A ¢ . * B ¢ * C / . . . . . 761 14.3 Kinematics of rolling and sliding . . . . . . . . . . . . . . . 767 Box 14.5 The Sturmey-Archer hub . . . . . . . . . . . . . 770 14.4 Mechanics of contact . . . . . . . . . . . . . . . . . . . . . 780 14.5 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 798 15 Kinematics using time-varying basis vectors 820 Here is a second take on the kinematics of particle motion but now using base vectors which change with time. The discussion of polar coordinates started in Chapter 13 is completed here. Path coordinates, where one base vector is parellel to the velocity and the others orthogonal to that, are introduced. The challenging topic of kinematics of relative motion is in two stages: first using rotating base vectors connected to a moving rigid object and then using the more abstract notation associated with frame-dependent differentiation and the famous “five term acceleration formula.” 15.1 Polar coordinates and path coordinates . . . . . . . . . . . . 821 15.2 Rotating frames and their base vectors . . . . . . . . . . . . 836 Box 15.1 The P * Q formula . . . . . . . . . . . . . . . . . . 845 15.3 General formulas for * v and * a . . . . . . . . . . . . . . . . . 850 Box 15.2 Moving frames and polar coordinates . . . . . . 856 15.4 Kinematics of 2-D mechanisms . . . . . . . . . . . . . . . . 862 15.5 Advanced kinematics of planar motion . . . . . . . . . . . . 875 Box 15.3 Skates, wheels and non-holonomic constraints . . 877 16 Constrained particles and rigid objects 888 The dynamics of particles and rigid bodies is studied using the relativemotion kinematics ideas from chapter 15. This is the capstone chapter for a two-dimensional dynamics course. After this chapter a good student should be able to navigate through and use most of the skills in the concept map inside the back cover. 16.1 Mechanics of a constrained particle . . . . . . . . . . . . . . 890 Box 16.1 Some brachistochrone curiosities . . . . . . . . . 896 16.2 One-degree-of-freedom 2-D mechanisms . . . . . . . . . . . 912 Box 16.2 Ideal constraints and workless constraints . . . . 913 Box 16.3 1 DOF systems oscillate at E P minima . . . . . . 918 16.3 Multi-degree-of-freedom 2-D mechanisms . . . . . . . . . . 926 Appendices 956 A Units & Center of mass theorems 956 Some things that are important, but don’t fit in the flow of a homeworkdriven course. First, issues related to units and dimensions, most importantly that a quantity is the product of a number and a unit. Thus units are part of a calculation. Some simple advice follows: a) balance units, b) carry units and c) check units. Rules for changing units also follow.
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